Sergiu Klainerman is an expert in PDEs and highly qualified to speak on both the Navier-Stokes problem and the future of mathematics. I was privileged to have him as my PhD advisor.
Despite the title, a good part of his essay addresses the formulation of the Navier-Stokes problem and whether it might have been a bit "too easy". If I could summarize in my own words, there are roughly three problems of increasing difficulty.
1. Forced: you're allowed to "stir the fluid" to try to produce an infinite vortex. This is what OpenAI accomplished, building on the works of others.
2. Unforced: no stirring allowed - can you construct a calmer fluid state at time t=0 so that an infinite vortex forms at a later time t=T? (Or equivalently at least stir the fluid and then let go before the infinite vortex forms.) Alternatively, prove that to be impossible.
3. General: understand and categorize conditions that lead to an infinite vortex, showing (most likely) that such a phenomenon arises only from contrived examples.
AI is pretty good at counterexamples. It will be interesting to see if AI can make progress on the general problem, which likely requires understanding of Navier-Stokes at a fundamentally deeper level.
The article seemed to fold in on itself; we aren't to worry because, ah, it turns out that taste was the most important thing in math all along (!?), and not ability or skill or outcome. Welcome to the humanities, I guess?
Clicking around: This is quite the political site. I see this sits alongside articles defending race realism a la Cofnas, and the site overall appears to be a clubhouse for academe's Bari Weiss style "woke right" flank.
I vaguely recall an old HN post in which someone asked roughly "Senior devs, what is your advice to younger devs?" What struck me most among the many responses was an observation that almost none of the advice was deeply technical. Nobody recommended what programming language to learn or dev tool to use. This was pre-LLMs, when devs actually had to write code themselves, but even then, senior devs saw a bigger picture.
A similar pattern applies in mathematics. We grow up thinking math is about computing integrals and software development is about implementing algorithms. But leading mathematicians think bigger. The author suggests that while AI can calculate integrals or logically test thousands of lemmas in a few hours, mathematicians still have to develop intuition for what problems to tackle next, and for large problems, what hard parts to focus on. I wouldn't call it taste - intuition can still end up wrong.
Sergiu Klainerman is an expert in PDEs and highly qualified to speak on both the Navier-Stokes problem and the future of mathematics. I was privileged to have him as my PhD advisor.
Despite the title, a good part of his essay addresses the formulation of the Navier-Stokes problem and whether it might have been a bit "too easy". If I could summarize in my own words, there are roughly three problems of increasing difficulty.
1. Forced: you're allowed to "stir the fluid" to try to produce an infinite vortex. This is what OpenAI accomplished, building on the works of others.
2. Unforced: no stirring allowed - can you construct a calmer fluid state at time t=0 so that an infinite vortex forms at a later time t=T? (Or equivalently at least stir the fluid and then let go before the infinite vortex forms.) Alternatively, prove that to be impossible.
3. General: understand and categorize conditions that lead to an infinite vortex, showing (most likely) that such a phenomenon arises only from contrived examples.
AI is pretty good at counterexamples. It will be interesting to see if AI can make progress on the general problem, which likely requires understanding of Navier-Stokes at a fundamentally deeper level.
No spam and spam activities will greatly hurt maths and all sciences.
I believe many OSS proejcts are also having problems with fact PRs and similar submissions.
All this spam is wasting the time of those custodians to accept and study real genuine submissions.
Unfortunate title when this is actually a better explanation of what OpenAI actually did and did not with Navier-Stokes
Maybe math evolves and humans are elevated from Ai? Hard to say!
The article seemed to fold in on itself; we aren't to worry because, ah, it turns out that taste was the most important thing in math all along (!?), and not ability or skill or outcome. Welcome to the humanities, I guess?
Clicking around: This is quite the political site. I see this sits alongside articles defending race realism a la Cofnas, and the site overall appears to be a clubhouse for academe's Bari Weiss style "woke right" flank.
So: firstly: yuck; secondly, nope.
I vaguely recall an old HN post in which someone asked roughly "Senior devs, what is your advice to younger devs?" What struck me most among the many responses was an observation that almost none of the advice was deeply technical. Nobody recommended what programming language to learn or dev tool to use. This was pre-LLMs, when devs actually had to write code themselves, but even then, senior devs saw a bigger picture.
A similar pattern applies in mathematics. We grow up thinking math is about computing integrals and software development is about implementing algorithms. But leading mathematicians think bigger. The author suggests that while AI can calculate integrals or logically test thousands of lemmas in a few hours, mathematicians still have to develop intuition for what problems to tackle next, and for large problems, what hard parts to focus on. I wouldn't call it taste - intuition can still end up wrong.